What is Helioseismology?

Helioseismology is the study of the interior of the Sun from
observations of the vibrations of its surface.

The Sun is nearly opaque to electromagnetic energy: it takes
about 170,000 years for radiation to get to the surface from the
core.

On the other hand, the Sun is essentially transparent to
neutrinos, and to acoustic waves. Using acoustic energy, we can
"see into the Sun" in a way that is quite similar to
using ultrasound to image the interior of the human body.

Oscillations of stars have been recognized since the late
1700s. The complicated pattern of the Sun's oscillation was first
observed in 1960 by Robert Leighton, Robert Noyes, and George
Simon.

The explanation of the pattern in terms of trapped acoustic
waves came in 1970-71 by Roger Ulrich, John Leibacher, and Robert
Stein. This explanation predicted certain detailed features of
the spectrum of solar oscillations that were confirmed by
observations made in 1975 by Franz Deubner.

The Sun is constantly vibrating in a superposition of acoustic
normal modes (like the patterns with which a guitar string
vibrates, but for a spherical body rather than a string). The
characteristic period of oscillation is about 5 minutes. It takes
on the order of a few hours for the energy to travel through the
Sun. The velocity amplitude of solar p-modes is about 1
cm/s; the relative brightness variation is about 10-7.

Mode lifetimes range from hours to months. Modes are typically
excited many times per lifetime.

The modes are thought to be excited by turbulence in the
convection zone.

About 107 distinct modes are thought to be excited;
of those, over 250,000 have been identified. It is thought that
within a few years, well over a million modes will have been
identified.

Helioseismology is rather like trying to understand how a
piano is built from the sounds that it makes when you drop it
down a flight of stairs.

Here are two movies of the motion of the surface of the Sun,
after filtering out solar rotation:GONG movie (from the GONG website) SOI Movie (from the Stanford SOHO-SOI website).

Here are some filtered, speeded-up sounds of the Sun,
processed by A. Kosovichev of the Stanford SOHO-SOI group. 40-day time
series of normal mode coefficients were combined and speeded up
by a factor of 42,000. One mode (l=1, n=20), three modes, all low-degree modes (l=0,
1, 2, 3).

One can look at the power in the motion as a function of
angular scale and temporal frequency, to obtain a
multi-dimensional power spectrum:

The observations of the Sun's oscillations are much better
than our observations of Earth's oscillations (low-frequency
geoseismology), partly because the oscillations go on constantly
(for the Earth, it takes a sizeable earthquake to excite
observable oscillations), and partly because we can observe
almost half the Sun at any time (on Earth, seismic stations are
scattered pretty sparsely). (The "almost" comes from
the fact that the solar limb is so noisy that it is usually
masked out of the data processing.)

To have a wave or a normal mode, the material must "push
back" when it is pushed, like a spring. The source of this
"restoring force" gives rise to different kinds of
oscillations and to modes that sample different parts of the Sun.
The observed modes of the Sun are surface gravity modes (f-modes:
the f stands for "fundamental") and p-modes,
for which the restoring force is pressure.

The particle motion associated with f-modes and p-modes
is essentially confined to a region outside the solar core (the
ray paths that the energy travels have turning depths above the
core), so those modes contain little information about the
deepest parts of the Sun.

In the core, negative buoyancy acts as a restoring force, so
the core supports "g-modes," for which gravity
is the restoring force.

g-modes sample the core well, but their associated
particle motion is essentially confined to the core. Their
amplitudes at the observable surface of the Sun are quite small,
and as yet, there is no convincing report that they have been
detected at all.

Figure from GONG website. See Gough, Leibacher, Scherrer, and Toomre, Science,
272, 1281-1283. The left figure shows ray paths
associated with two p modes (shallow is l=100, n=8;
deep is l=2, n=8); the right shows the ray path
associated with a g mode (l=5, n=10).

The patterns that individual modes make at the surface of the
Sun are quite similar to spherical harmonics Ylm,
which are the analogues of sines and cosines on the sphere. (They
are eigenfunctions of Laplaces's equation; they diagonalize any
operator that commutes with rigid-body rotation.)

To first order in the (small) rotation rate, the relationship
between the portion of the angular velocity distribution that is
symmetric w.r.t. the solar equator, and the odd (in m)
component of the splitting, is linear, if one pretends that the
eigenfunctions of the oscillations are known.

Figure from the GONG website. See Thompson, Toomre, Anderson, and 26 others,
1996. Science, 272, 1300-1305. The left three
figures show the sensitivity of some frequency splittings to
rotation of the Sun at different latitudes and depths. The
rightmost figure shows a combination of splitting sensitivities
designed to "target" the rotation at certain places
within the Sun.

Overall Goals of
Helioseismology

Learn about the composition, state, and
dynamics of the interior of the closest star, including
sunspots, the heliodynamo, and the solar cycle

Test and improve theories of stellar
evolution

Use the Sun as a physics laboratory to
study conditions unattainable on Earth (e.g.,
the neutrino problem, the equation of state, opacity)

Possible By-Products

Predict space weather by imaging the far
side of the Sun

Notable Successes of
Helioseismology

The agreement between the predicted spectra for ab
initio models and the observed spectra was truly remarkable,
but as observations improved (and error bars decreased), it
became clear that the "standard solar model" was wrong
in important ways.

Since the mid-1980s, many studies of solar rotation using
frequency splittings have shed doubt on dynamo models
that required rotation to be roughly constant on
cylinders in the convection zone.

Errors in opacity calculations of numerical
nuclear physicists

The "standard solar model" fit the estimates of
soundspeed better if the opacity at the base of the
convection zone was modified in an ad hoc way.
Checking with the physicists who produced the original
opacity figures led to the discovery that the bound-state
contribution of iron had been underestimated, as a result
of the hydrogenic approximation. That error led to a
10%-20% error in the opacity at the base of the
convection zone. Revised opacity calculations brought
theory in line with solar observations, and explained the
pulsation period ratios of Cepheid stars, previously a
mystery.

Progress in the solar neutrino problem.

Measurements of the solar neutrino flux over 25 years
lower than predicted by nuclear physics in conjunction
with stellar evolutionary models. Not yet clear whether
the problem is with nuclear physics (e.g.,
neutrinos might have mass) or with the theory of stellar
evolution. Observations of the modes that do probe the
core (to some extent) make low Helium abundance in the
core an implausible explanation of the solar neutrino
deficit.

Recent Announcements

"Plasma rivers" in the Sun.
The SOHO-SOI/MDI team announced last year the discovery of
"plasma rivers" in the Sun, (press
release) where the flow of solar plasma is about 10% faster
than the surrounding material. This was one of the top 10 NASA
stories for 1997.

Two Current
Helioseismic Experiments

There are a number of experiments in different
countries, ranging from networks that view the Sun as a star
(spatially unresolved) to very high resolution single stations.
I'll just describe the two I am affiliated with. These are the
experiments with high spatial resolution and the highest duty
cycles. Duty cycle is quite important, because gaps in the
observation series result in spurious artifacts in the estimates
of the spectrum of solar oscillations.

The Global Oscillations Network Group (GONG) is a
6-station ground-based network funded by the NSF.

The Solar and Heliospheric Observer Solar
Oscillations Investigation (SOHO-SOI) is a satellite-based
experiment funded by NASA and ESA.

Both try to get essentially continuous
observations, GONG, by having a globe-spanning network on which
the Sun never sets, and SOHO-SOI by observing from space (a halo
orbit around the L1 Lagrange point).

These cubes are the heart of the Michelson
Doppler Interferometers used both by GONG and SOHO-SOI/MDI to
determine the pattern of motion of the solar surface. Fourier
tachymeters for measuring solar oscillations were first developed
by Tim Brown (HAO/NCAR) in the 1980's. The basic idea is to use a
sequence of filters to isolate an absorption line originating in
the mid-photosphere (Ni I, 676.8nm), then use tunable
interferometers to measure the intensity at several frequencies
in a small band; these can in turn be converted to estimates of
the Doppler shift in the absorption line, and hence to a
velocity. This is done in each pixel of the image (in a CCD
camera essentially identical to ones used in video cameras),
resulting in a spatially resolved image of the velocity of the
Sun's surface. Both GONG and SOHO-SOI/MDI produce one such image
per minute. The SOHO-SOI/MDI camera has 1024 by 1024 pixels; the
GONG cameras have 242 by 256 pixels. The overall data collection
rate for GONG is about 38Gb/month. The basic analysis unit for
studying oscillations is a time series of 36 or 72 days of such
time series (51,840 or 103,680 images).

This is a velocity image after
filtering to remove the Sun's rotation.

The
GONG Data Pipeline

Figure from the GONG website. See Harvey, Hill, Hubbard, and 14 others, 1996,
Science, 272, 1284-1286. Schematic of the data processing
flow: time series of images like that on the left are each
decomposed into spherical harmonics, giving many time series of
spherical harmonics like that in the middle. The spectrum of each
of those time series is estimated, as a function of frequency.
The acoustic power as a function of the frequency and angular
wavenumber can then be estimated, as on the right.

From GONG website. See Hill, Stark, Stebbins, and 23 others,
1996. Science, 272, 1292-1295. Examples of
spectra well and poorly fitted by the parametric models used to
estimate the frequency, amplitude, linewidth, and background
power of solar modes.

Figure from the GONG website. See Hill, Stark, Stebbins, and 23 others,
1996. Science, 272, 1292-1295. Central
frequencies of modes as a function of l, averaged over m,
with formal error bars magnified by 200.

Asterisks (*)
indicate steps that could be improved or possibly eliminated by
better statistical methodology.

Inversion

The two principal kinds of information one can
extract from helioseismic data are spatial averages of the speed
with which seismic waves travel in the Sun, and spatial averages
of the speed with which parts of the Sun are moving relative to
other parts (because the seismic waves are advected with the
material). Until a few years ago, the features studied were
global, and the studies were based on normal mode frequency
estimates; more recently, scientists have started using
"local area helioseismology" to image local features of
the Sun, such as the flow beneath sunspots, using time series of
images more directly.

These inference (inverse) problems are quite
difficult, but they share mathematical and statistical structure
with inverse problems in may fields, including geophysics and
medical imaging.

Figure from the GONG website. See Thompson, Toomre, Anderson, and 26 others,
1996. Science, 272, 1300-1305. Left panel is an
estimate of the average rotation as a function of latitude and
radius using an approximate separation of latitude and radius in
the forward problem. Right panel is a regularized least-squares
model in a tensor-product basis, with second-derivative smoothing
and an arbitrary choice of Lagrange multiplier.

Figure from the GONG website. See Thompson, Toomre, Anderson, and 26 others,
1996. Science, 272, 1300-1305. Sections through
the two-dimensional images above, at three latitudes. Shading is
±1SD(nominal).

Statistical
Questions and Opportunities

Data Reduction and Calibration: how best to account for
instrumental drifts, changes in lens cleanliness, etc.